Conservation Of Kinetic Energy Research Articles

On a symmetry-preserving unconditionally stable projection method on collocated unstructured grids for incompressible flows

This work establishes the conditions for having a symmetry-preserving unconditionally stable projection method for incompressible flows, such as PISO or Fractional Step Method (FSM), on collocated unstructured grids based on a compact stencil Poisson equation. Its formulation is based on preserving the underlying symmetries of the differential operators. In addition, a general theorem for these types of projection methods will be proven. To establish an unconditionally stable method, this theorem gives the mathematical requirements for the operators and the geometrical conditions a mesh must satisfy, even in cases where the mesh is highly distorted. This will be proven both theoretically and numerically. Conservation of (global) kinetic energy is also a key feature in simulations. Within this context, two canonical cases are tested, a turbulent channel flow at Reτ=395 to show the robustness of the method, and an air-filled differentially heated cavity at Ra=1010 (based on the cavity height), to show that the (artificial) kinetic energy error introduced by the pressure is negative and very small compared with the physical dissipation.

Energy-conserving finite difference scheme based on velocity interpolation applicable to unsteady flows using collocated grids

The collocation method uses the Rhie–Chow scheme to find the cell interface velocity by pressure-weighted interpolation. The accuracy of this interpolation method in unsteady flows has not been fully clarified. This study constructs a finite difference scheme for incompressible fluids using a collocated grid in a general curvilinear coordinate system. The velocity at the cell interface is determined by weighted interpolation based on the pressure difference to prevent pressure oscillations. The Poisson equation for the pressure correction value is solved with the cross-derivative term omitted to improve calculation efficiency. In addition, simultaneous relaxation of velocity and pressure is applied to improve convergence. Even without the cross-derivative term, calculations can be stably performed, and convergent solutions are obtained. In unsteady inviscid flow, the conservation of kinetic energy is excellent even in a non-orthogonal grid, and the calculation result has second-order accuracy to time. In viscous analysis at a high Reynolds number, the error decreases compared with that of the Rhie–Chow interpolation method. The present numerical scheme improves calculation accuracy in unsteady flows. The possibility of applying this computational method to high Reynolds number flows is demonstrated through several analyses.

Generalized eddy‐diffusivity mass‐flux formulation for the parametrization of atmospheric convection and turbulence

AbstractThe atmospheric convection is a phenomenon that has a length scale smaller than the resolution at which the state‐of‐the‐art general circulation numerical models are running, and thus the convection needs to be parametrized. In this work, a theoretical formulation for the parametrization of subgrid‐scale convection based on subgrid averaging that represents a generalized eddy‐diffusivity mass‐flux formulation is presented. The subgrid fluxes are derived by considering a decomposition of subgrid variables into convective and turbulent variables, and by assuming that the convection is modeled by round convective plumes with generic radial profiles. The main difference between our formulation and the mass‐flux formulations is that the condition of a very small fractional area occupied by the convection is replaced with the condition that the convective vertical velocity goes to the mean state far from the updraft region of the plume. The plume model can also be generalized by considering that the convective elements are energy‐consistent plumes, governed by the conservation of mass, momentum, kinetic energy, and buoyancy, which provides a physical‐based closure for the entrainment. Therefore, the closing problem of our formulation consists of the specification of the convective radial profiles and the boundary conditions at the initial vertical level. Furthermore, our formulation allows one to consider more realistic profiles for the convective variables, making the formulation suitable for the parametrization of atmospheric convection at the convective gray zone. This aspect is discussed in the last part of this work, where it is showed how the formulation can be implemented at any resolution. Moreover, in the present framework, no distinct assumptions are required for each convective type, thus facilitating the parametrization of convection in a unified way.

A simple analytical model for turbulent kinetic energy dissipation for self-similar round turbulent jets

This work presents a simple analytical model for the streamwise and radial variations of turbulent kinetic energy dissipation in an incompressible round turbulent jet. The key assumptions in the model are: similarity in the axial velocity profile with a Gaussian shape, axisymmetric flow and the dominance of radial derivatives of the mean velocity over axial direction derivatives (similar to boundary layer theory). Initially, a simplified eddy-viscosity relation for turbulent stresses is derived using the algebraic stress model by Gatski & Speziale (J. Fluid Mech., vol. 254, 1993, pp. 59–78). Subsequently, with this eddy-viscosity relation, the relation for variations of turbulent kinetic energy dissipation is formulated using the conservation of turbulent kinetic energy. To extract the necessary constants of the model, experimental velocity statistics for round jets are obtained through particle image velocimetry measurements. The experimental results of the mean entrainment coefficient for turbulent jets are also analysed. When comparing the radial variation of turbulent kinetic energy dissipation from the model with experimental results at Reynolds number $1.4\times 10^5$ and numerical results at Reynolds number $1200$ from the available literature, we observe a maximum error of $10\,\%$ and $15\,\%$ , respectively. Finally, using the validated model, we analyse the impact of mean velocity evolution parameters on the behaviour of turbulent kinetic energy dissipation and discuss its potential significance in future studies.

Non-linear response of kinetic energy in an inviscid Taylor-Green flow obtained by OpenFOAM-LES with respect to time increments

This study investigates the conservation error of kinetic energy with respect to time increment using OpenFOAM LES. The energy conservation error is verified using an inviscid Taylor-Green vortex and the effect on energy conservation is checked by varying the time increment by a factor of 30. Differences in the response of turbulent energy conservation between large and small time increments are also verified. Higher order accuracy for time increments is expected when the kinetic energy conservation accuracy is high, however the actual behaviour in OpenFOAM needs to be investigated: as the implicit time integration method is used in OpenFOAM, a non-linear response is expected. As a result, it is evident that kinetic energy is not fully conserved in the inviscid flow of OpenFOAM. The kinetic energy distribution is similar for different time increments, with a different relationship between the conservation error observed in the range of large and small time increments.

Energy conservation uncertainly due to the use of an implicit time integration method clarified by a multiplexed TGV inviscid flow

This study attempts to elucidate the conservation errors induced by the implicit time integration method within a multiscale flow. Using the multiplexed Taylor analytical solution as the initial field, three distinct multiscale turbulence fields were formulated, each characterised by different wavenumber attributes. A two-dimensional inviscid flow field, replicated in a computational domain with a side of 2π and subject to periodic boundary conditions, was used to illuminate kinetic energy conservation errors. The implicit time integration method - specifically the second-order accurate Crank-Nicolson method - was compared with explicit methods, such as the second-order accurate Adams-Bashforth and third-order accurate Runge-Kutta methods. The study also examines the differences between the analytical Taylor solution and a random flow field, particularly in their satisfaction of the Navier-Stokes equations and wavenumber composition, while highlighting the need for preliminary analysis for random flow fields prior to validation calculations.

Multiscaled inviscid Taylor-Green vortex flow for examining energy conservation error in incompressible flows

This study investigates kinetic energy conservation errors in incompressible flows using multiplexed inviscid Taylor-Green vortices. Fourth-order precision Runge-Kutta methods, specifically five- and six-step methods, were employed and an inviscid two-dimensional periodic flow field was used as a test case. This method allows the energy conservation error to be suppressed to a very low level. Recent previous studies have dealt with turbulence generation using a multi-scale turbulence grid. In this study, the Taylor-Green vortex of the analytical solution is multiplexed to generate a flow field with a number of wavenumbers. Visualisation results of the initial field and the energy distribution over time were obtained. It was observed that the errors in the time evolution of the fluid energy were extremely small, and that the errors actually increased with time. However, the influence of the time step range was limited and the analytical and numerical solutions were in general agreement. As a result, it was confirmed that the inviscid Taylor-Green vortex is effective in examining the energy conservation error for incompressible flows.

Assessing OpenFOAM-based Large-eddy Simulation Using Decay Characteristics of An Isotropic Taylor-green Vortex

The aim of this study is to investigate OpenFOAM using isotropic TGV flows. The main focus is on the establishment of energy conservation laws, in particular using inviscid fields to assess energy constancy; Large-Eddy Simulation (LES) is applied and the results are compared with those obtained using spectral methods; unlike previous studies of OpenFOAM, this study uses an unsteady turbulent TGV flow. The computational methods used are a central difference method, a PISO algorithm and a Smagorinsky model. The study revealed errors in the conservation of kinetic energy in the OpenFOAM analysis. In particular, the energy decay was significant at high spatial resolution, confirming the different turbulence energy decay characteristics of DNS and OpenFOAM. However, when the model constant was 0.6, the decay characteristics were shown to be in good agreement with DNS.

Studying on the upper limit of the coarse-grained DEM model for simulating mixing processes in a rolling drum with complex wall boundary

Compared with the original Discrete Element Method, the coarse-grained DEM significantly improves simulation efficiency. In CG DEM, the coarse-grained ratio is defined as the diameter of the coarse-grained particle divided by the diameter of the original particle. When the upper limit of coarse-grained ratio is used, the simulation is the most efficient. The upper value for simulating mixing process of spherical fuel particles was determined using a multi-level methodology. The adequacy of the upper value of the coarse-grained ratio was assessed by comparing the mixing characteristics and kinetic properties of the coarse-grained particle systems with those of the original particle systems in a rotating drum with blades. The impact of rotation speed and particle number on the upper value was examined and discussed. It was observed that the kinetic energy increases with an increase in the coarse-grained ratio, limiting the efficiency improvement. To ensure kinetic energy conservation, a scaling law for the restitution coefficient was proposed. Combined with the Reduced Particle Stiffness method, the simulation time of the coarse-grained case is only 1/3030 that of the original case (with a particle number of 2.3 million) when the upper limit of the coarse-grained ratio was used (12.0).

Energy-conserving hyper-reduction and temporal localization for reduced order models of the incompressible Navier-Stokes equations

A novel hyper-reduction method is proposed that conserves kinetic energy and momentum for reduced order models of the incompressible Navier-Stokes equations. The main advantage of conservation of kinetic energy is that it endows the hyper-reduced order model (hROM) with a nonlinear stability property. The new method poses the discrete empirical interpolation method (DEIM) as a minimization problem and subsequently imposes constraints to conserve kinetic energy. Two methods are proposed to improve the robustness of the new method against error accumulation: oversampling and Mahalanobis regularization. Mahalanobis regularization has the benefit of not requiring additional measurement points. Furthermore, a novel method is proposed to perform energy- and momentum-conserving temporal localization with the principle interval decomposition: new interface conditions are derived such that energy and momentum are conserved for a full time-integration instead of only during separate intervals. The performance of the new energy- and momentum-conserving hyper-reduction methods and the energy- and momentum-conserving temporal localization method is analysed using three convection-dominated test cases; a shear-layer roll-up, two-dimensional homogeneous isotropic turbulence and a time-periodic inviscid flow consisting of a vortex in a uniform background flow. Our main finding is that energy conservation in combination with oversampling or regularization leads to a robust method with excellent long time stability properties. When any of these two ingredients is missing, accuracy and/or stability is significantly impaired.

An Off-Center, Elastic Collision: Conservation of Kinetic Energy Without a Perfect Coefficient of Restitution

An Off-Center, Elastic Collision: Conservation of Kinetic Energy Without a Perfect Coefficient of Restitution

Ragone plots of material-based hydrogen storage systems

This paper presents an analytical assessment of the energy–power relationship for different material-based hydrogen storage systems, namely Metal Hydrides (MHs) and Liquid Organic Hydrogen Carriers (LOHCs). Storage systems are subjected to continuous flow discharge processes through suitable control systems to meet constant specific power demands of end users. By means of reasonable assumptions, analytical expressions of the time-dependent degree of hydrogenation (representing the state of charge) are obtained to find the amount of hydrogen discharged (delivered energy) as a function of flow rate (required power). The results are first presented in the form of dimensionless Ragone plots to highlight the dependence of the amount of discharged hydrogen on the required mass flow rate. Numerical examples are presented for a couple of illustrative systems. Moreover, these analytical expressions are shown to produce very similar results to those obtained with the solution of a dynamic model, including, beyond the kinetic equation, mass and energy conservation applied to the reactor. The results show a significant impact of power demand on the released hydrogen for most systems, similar to that of capacitors, due to the dependence of the rate of reaction on the degree of hydrogenation: as a consequence, the amount of energy that can be delivered to an end user decreases substantially with an increase in the required power, resulting in a poor utilisation factor. Based on these results, MHs exhibit almost first-order kinetics and can sustain efficient discharge with theoretical specific power up to 2kW/kg (rate of chemical energy delivered per unit mass of active substance), corresponding to a discharge duration of the order of 0.25h; some LOHCs are limited by second-order kinetics and the specific power should be lower than 1kW/kg, with discharge durations that must be above 2hour, to ensure effective utilisation of stored hydrogen.

Theory of gyroscopic effects for rotating objects

Scientists began to study gyroscopic effects at the time of the Industrial Revolution. Famous mathematician L. Euler described only one gyroscopic effect, which is the precession torque that does not explain other ones. Since those times, scientists could not explain the physics of gyroscopic effects, Recent studies and the method of causal investigatory dependency demonstrated, that the nature of gyroscopic effects turned out that be more sophisticated than contemplated by researchers. The external torque acting on the spinning objects generates the system of the eight inertial torques and their interrelated motions around axes presented in the 3D coordinate system. The interrelated torques and motions of the spinning disc were described by mathematical models, and validated by practical tests that explain the physics of the gyroscopic effects based on the kinetic energy conservation law. The inertial torques generated by the centrifugal, and Coriolis forces, the change in the angular momentum, and the dependent motions of the spinning object around axes constitute the fundamental principles of the gyroscope theory. The derived gyroscopic theory opened a new chapter in the dynamics of rotating objects of classical mechanics that should be presented in all word encyclopedias.

Newtonian cosmology from quantum corrected Newtonian potential

We study the Newtonian cosmology taking into account the leading classical and quantum corrections of order O(G2) in the Newtonian potential. We first derive the modified Friedmann equations starting from the non-relativistic conservation of kinetic energy and potential energy for an infinitesimal mass. We then consider the leading classical correction term and the quantum correction term in the Newtonian potential for deriving the Friedmann equation, however, the quantum correction term is too small and hence does not contribute in the physical results. We then investigate the difference in scale factor with the usual scale factor for various matters like radiation, dust and cosmological constant by considering the corrections in the Newtonian potential. We observe that the evolution of the universe is similar for radiation and dust cases at late times. The cosmological constant case shows a steep increase in the scale factor compared to the other cases. We also note that the universe may have a bounce in the case of radiation depending on the sign of the coefficient of the leading classical correction.

Drift-ordered fluid vorticity equation with energy consistency

Although drift-ordered fluid models are widely applied in tokamak edge turbulence simulations, the models used are acknowledged not to conserve energy or even electrical charge. The present paper aims to remove many of the existing pitfalls in drift-fluid models, however, with the objective of finding a solution simple enough to be implemented in numerical applications. Our main result is an improved version of the drift-Braginskii equations involving a generalized vorticity function. In the new drift-Braginskii system, the quasi-neutrality condition translates into a transport equation for a generalized vorticity, expressed in conservation form, and related to the total mass-weighted circulation. It is found that kinetic energy conservation can be achieved if the polarization flow is defined recursively. The resulting model conserves the kinetic energy associated with E×B and diamagnetic flows and retains the associated perpendicular kinetic energy flux.

Non-linear wave propagation in a weakly compressible Kelvin-Voigt liquid containing bubbly clusters

The effect of bubble-bubble interaction on wave propagation in homogeneous weakly compressible viscoelastic bubbly flow is investigated using the reductive perturbation method. The bubble dynamics equation is derived using the kinetic energy conservation approach. The bubble dynamics and mixture equations are coupled with the equation of state for gas to investigate the shock wave propagation phenomenon in the mixture. A two-dimensional Korteweg-de VriesBurger (KdVB) equation in terms of a pressure profile is derived. It is found that the bubble-bubble interaction has no effect when using the parameters under our consideration.

Diagnosing Mental Models of Undergraduate Student and Physics Teachers: Study Case in the Momentum and Energy Conservation Principles Using Newton's Cradle

The Newton’s Cradle comprises a series of pendulums often used in physics classrooms to demonstrate the principles of conservation of momentum and kinetic energy in elastic collisions. By utilizing open-ended questions and several Newton’s cradle scenarios, this study aims to clarify how the Newton’s cradle can be used for assessing the mental models of the principles of conservation of momentum and energy. Interviews with 18 college students and five physics professors each lasted 30 to 45 minutes. Firstly, they were asked to explain how the Newton’s cradle works. Next, a scenario started with three balls with equal masses where one collides with the other two, they were asked to explain if the possible outcomes for a collision with initial momentum = mv1 + 0 + 0 and the final momentum = (-mv1/3) + (2mv1/3) + (2mv1/3). With a five-ball Newton’s Cradle, students were asked to explain the outcomes when 1) one collides with the other four, 2) two collide with the other three, 3) three collide with the other two, and 4) four collide with the last one. Their drawings and explanations dealing with the Newton’s cradle during the interviews were analyzed for their mental models and categorized into patterns. The results showed that both university students and physics teachers hold misinformed conceptions about conservation of momentum and energy and have unsound mental models in the context of Newton’s Cradle. We also found that they did not recognize that sound and heat or lack of exact alignment of the balls are factors accountable for loss of energy causing the Newton’s Cradle to depart from idealized situation. Based on the interview data, teaching principles of the conservation of momentum and energy through use of Newton’s cradle are deficient and curriculum review is suggested.

Global and local conservation of mass, momentum and kinetic energy in the simulation of compressible flow

The spatial discretization of convective terms in compressible flow equations is studied from an abstract viewpoint, for finite-difference methods and finite-volume type formulations with cell-centered numerical fluxes. General conditions are sought for the local and global conservation of primary (mass and momentum) and secondary (kinetic energy) invariants on Cartesian meshes. The analysis, based on a matrix approach, shows that sharp criteria for global and local conservation can be obtained and that in many cases these two concepts are equivalent. Explicit numerical fluxes are derived in all finite-difference formulations for which global conservation is guaranteed, even for non-uniform Cartesian meshes. The treatment reveals also an intimate relation between conservative finite-difference formulations and cell-centered finite-volume type approaches. This analogy suggests the design of wider classes of finite-difference discretizations locally preserving primary and secondary invariants.

Validation on the mean friction velocity of an atmospheric boundary layer flow reproduced by large-eddy simulation in terms of kinetic energy conservation

This study presents a validation of large-eddy simulation to reproduce normalized mean friction velocity in an atmospheric boundary layer flow in terms of kinetic energy conservation uncertainty. A primary finding of this study is that the normalised mean friction velocity of the atmospheric boundary layer is suggested to be insensitive to kinetic energy conservation errors. The present study approaches the sensitivity of the normalized friction velocity, which is one of the most fundamental statistics in the atmospheric boundary layer, to the uncertainty. The nearly complete conservation of kinetic energy in the present numerical framework is verified in an inviscid homogeneous isotropic fluctuation field. Then, the present analysis is applied to reproducing turbulent channel flows as direct numerical simulation and large-eddy simulation based on the previous studies before the present work analyses the present atmospheric boundary layer. The present analysis is then used to analyze the normalized friction velocity in the atmospheric boundary layer. The values of the friction velocity obtained in the previous study are in good agreement with that of the present study.

On Maximally Mixed Equilibria of Two-Dimensional Perfect Fluids

The vorticity of a two-dimensional perfect (incompressible and inviscid) fluid is transported by its area preserving flow. Given an initial vorticity distribution $\omega_0$, predicting the long time behavior which can persist is an issue of fundamental importance. In the infinite time limit, some irreversible mixing of $\omega_0$ can occur. Since kinetic energy $\mathsf{E}$ is conserved, not all the mixed states are relevant and it is natural to consider only the ones with energy $\mathsf{E}_0$ corresponding to $\omega_0$. The set of said vorticity fields, denoted by $\overline{\mathcal{O}_{\omega_0}}^*\cap \{ {\mathsf E}= {\mathsf E}_0\}$, contains all the possible end states of the fluid motion. A. Shnirelman introduced the concept of maximally mixed states (any further mixing would necessarily change their energy), and proved they are perfect fluid equilibria. We offer a new perspective on this theory by showing that any minimizer of any strictly convex Casimir in $\overline{\mathcal{O}_{\omega_0}}^*\cap \{ {\mathsf E}= {\mathsf E}_0\}$ is maximally mixed, as well as discuss its relation to classical statistical hydrodynamics theories. Thus, (weak) convergence to equilibrium cannot be excluded solely on the grounds of vorticity transport and conservation of kinetic energy. On the other hand, on domains with symmetry (e.g. straight channel or annulus), we exploit all the conserved quantities and the characterizations of $\overline{\mathcal{O}_{\omega_0}}^*\cap \{ {\mathsf E}= {\mathsf E}_0\}$ to give examples of open sets of initial data which can be arbitrarily close to any shear or radial flow in $L^1$ of vorticity but do not weakly converge to them in the long time limit.